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Some new Farkas-type results for inequality systems with DC functions. (English) Zbl 1182.90071
The authors study the DC (difference of convex functions) optimization problem (P): $\text{Inf} (g(x)-h(x))$ s.t. $x\in X$, $g_i(x)-h_i(x) \le 0$ $(i=1, \dots, m)$, where $x\subseteq\Bbb R^n$ is a non-empty convex set, $g, h:\Bbb R^n\rightarrow \overline{\Bbb R}$ are two proper convex functions and $g_i, h_i:\Bbb R^n\rightarrow \overline{\Bbb R}$ $(i=1, \dots, m)$ are proper convex functions such that $\bigcap_{i=1}^m r_i (\text{dom}(g_i))\cap r_i(\text{dom}(g))\cap r_i (X)\neq \phi$. It is assumed that $h$ is lower semicontinuous and $h_i$ $(i=1, \ldots, m)$ are subdifferentiable on the feasible set of (P). A Fenchel-Lagrange dual problem for (P) is constructed and using the technique of {\it J.-E. Martinez-Legaz} and {\it M. Volle} [J. Math. Anal. Appl. 237, No. 2, 657--671 (1999; Zbl 0946.90064)] a dual problem to (P) is associated. The authors then use these results to derive Farkas-type results for inequality systems involving DC functions. It is claimed that some equivalent formulations of known results are obtained.

90C26Nonconvex programming, global optimization
90C46Optimality conditions, duality
Full Text: DOI
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